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Solving Exponential Equations

Exponential equations are equations in which variables occur as exponents.

For example, exponential equations are in the form a x = b y .

To solve exponential equations with same base, use the property of equality of exponential functions .

If b is a positive number other than 1 , then b x = b y if and only if x = y . In other words, if the bases are the same, then the exponents must be equal.

Example 1:

Solve the equation 4 2 x 1 = 64 .

 Note that the bases are not the same. But we can rewrite 64 as a base of 4 .

We know that, 4 3 = 64 .

Rewrite 64 as 4 3 so each side has the same base.

4 2 x 1 = 4 3

By the property of equality of exponential functions, if the bases are the same, then the exponents must be equal.

2 x 1 = 3

Add 1 to each side.

2 x 1 + 1 = 3 + 1 2 x = 4

Divide each side by 2 .

2 x 2 = 4 2 x = 2

 

Note:

If the bases are not same, then use logarithms to solve the exponential equations. See Solving Exponential Equations using Logarithms .

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